VARIATION FORMULAS OF SOLUTION FOR A CONTROLLED FUNCTIONAL-DIFFERENTIAL EQUATION CONSIDERING DELAY
PERTURBATION
T. TADUMADZE1, A. NACHAOUI2 §
Abstract. Variation formulas of solution are proved for a controlled non-linear functional-differential equation with constant delay and the continuous initial condition. In this pa-per, the essential novelty is the effect of delay perturbation in the variation formulas.The continuity of the initial condition means that the values of the initial function and the trajectory always coincide at the initial moment.
Keywords: Controlled delay functional-differential equation; variation formula of solu-tion; effect of delay perturbasolu-tion; continuous initial condition.
AMS Subject Classification: 34K99
1. Introduction
Linear representation of the main part of the increment of a solution of an equation with respect to perturbations is called the variation formula of solution (variation formula).The variation formula allows one to construct an approximate solution of the perturbed equa-tion in an analytical form on the one hand, and in the theory of optimal control plays the basic role in proving the necessary conditions of optimality [1-3,6,7,10], on the other. Vari-ation formulas for various classes of functional-differential equVari-ations without perturbVari-ation of delay are given in [2-5,7,8,10,11]. Here we are interested in the variation formulas for the controlled delay functional-differential equation
˙
x(t) =f(t, x(t), x(t−τ0), u0(t)) with the continuous initial condition
x(t) =ϕ0(t), t∈[t00−τ0, t00]
under perturbations of initial moment t00, delay parameter τ0, initial function ϕ0(t) and control functionu0(t). In this paper, the essential novelty is the effect of perturbation of delay τ0 in the variation formulas (see c2). The variation formula in a neighborhood of
1 Tbilisi State University, Department of Mathematics, 2, University St., 0186, Tbilisi, Georgia,
e-mail: [email protected]
2University of Nantes/CNRS UMR 6629, Jean Leray Laboratory of Mathematics , 2 rue de Houssiniere,
B.P. 92208, 44322 Nantes, France,
e-mail: [email protected] § Manuscript received 03 May 2011.
TWMS Journal of Applied and Engineering Mathematics Vol.1 No.1 c°I¸sık University, Department of Mathematics 2011; all rights reserved.
the right end of the main interval (so called local variation formula) for the differential equation
˙
x(t) =f0(t, x(t), x(t−τ0)) (1.1) with the initial condition
x(t) =ϕ0(t), t∈[t00−τ0, t00), x(t00) =x00,
when perturbation of initial data, delay and right-hand side of the equation (1.1) occurs is proved in [12]. It is important to note that the variation formula which is proved in the present work doesn’t follows from the formula proved in [12] (seec6).
2. Notation and auxiliary assertions
Let Rn
x be the n-dimensional vector space of points x = (x1, ..., xn)T, where T means transpose; suppose thatO ⊂Rn
x andV ⊂Rruare open sets. Let then-dimensional function f(t, x, y, u) satisfies the following conditions: for almost all t ∈ J = [a, b], the function f(t,·) :O2×V →Rn
x is continuously differentiable; for any (x, y, u)∈O2×V,the functions f(t, x, y, u), fx(·), fy(·), fu(·) are measurable on J; for arbitrary compacts K ⊂O, U ⊂V there exists a functionmK,U(·) ∈ L(J,[0,∞)), such that for any (x, y, u) ∈ K2×U and for almost allt∈J the following inequality is fulfilled
|f(t, x, y, u)|+|fx(·)|+|fy(·)|+|fu(·)|≤mK,U(t).
Further, let 0 < τ1 < τ2 be given numbers; Eϕ be the space of continuous functions ϕ:J1 → Rnx, where J1 = [ˆτ , b],τˆ = a−τ2; Φ ={ϕ ∈Eϕ : ϕ(t) ∈O, t ∈ J1} be a set of initial functions; let Eu be the space of bounded measurable functions u : J → Rur and Ω ={u∈Eu :clu(J)⊂V} be a set of control functions, whereu(J) ={u(t) :t∈J} and clu(J) is the closer of the set u(J).
To each element µ= (t0, τ, ϕ, u)∈Λ = (a, b)×(τ1, τ2)×Φ×Ω we assign the controlled delay functional-differential equation
˙
x(t) =f(t, x(t), x(t−τ), u(t)) (2.1) with the initial condition
x(t) =ϕ(t), t∈[ˆτ , t0]. (2.2) The condition (2.2) is said to be continuous initial condition since alwaysx(t0) =ϕ(t0).
Definition 2.1. Let µ = (t0, τ, ϕ, u) ∈ Λ. A functionx(t) =x(t;µ) ∈O, t ∈[ˆτ , t1], t1 ∈ (t0, b), is called a solution of equation (2.1) with the initial condition (2.2) or a solution corresponding to µ and defined on the interval [ˆτ , t1] if it satisfies condition (2.2) and is absolutely continuous on the interval [t0, t1] and satisfies equation (2.1) almost everywhere on [t0, t1].
Letµ0 = (t00, τ0, ϕ0, u0)∈Λ be a fixed element. In the spaceEµ=R1t0×R
1
τ×Eϕ×Eu we introduce the set of variations:
V ={δµ= (δt0, δτ, δϕ, δu)∈Eµ−µ0 :|δt0 |≤α,|δτ |≤α, δϕ=
k
X
i=1
λiδϕi, δu= k
X
i=1
λiδui,|λi|≤α, i= 1, k}, (2.3)
Lemma 2.1. Let x0(t) be the solution corresponding to µ0 = (t00, τ0, ϕ0, u0) ∈ Λ and
defined on [ˆτ , t10], t10 ∈ (t00, b) and let K0 ⊂ O and U0 ⊂V be compact sets containing
neighborhoods of sets ϕ0(J1) ∪x0([t00, t10]) and clu0(J), respectively. Then there exist numbers ε1 >0 and δ1>0 such that, for any(ε, δµ)∈[0, ε1]×V, we have µ0+εδµ∈Λ.
In addition, a solutionx(t;µ0+εδµ) defined on the interval [ˆτ , t10+δ1]⊂J1 corresponds
to this element. Moreover, (
x(t;µ0+εδµ)∈K0, t∈[ˆτ , t10+δ1], u0(t) +εδu(t)∈U0, t∈J.
(2.4)
This lemma is a result of Theorem 5.3 in [9, p.111].
Remark 2.1. Due to the uniqueness, the solutionx(t;µ0) is a continuation of the solution x0(t) on the interval [ˆτ , t10+δ1].Therefore, in the sequel the solution x0(t) is assumed to be defined on the interval [ˆτ , t10+δ1].
Lemma 2.1 allows one to define the increment of the solution x0(t) =x(t;µ0) :
(
∆x(t) = ∆x(t;εδµ) =x(t;µ0+εδµ)−x0(t), (t, ε, δµ)∈[ˆτ , t10+δ1]×[0, ε1]×V.
(2.5)
Lemma 2.2. Let the following conditions hold:
2.1. the functionϕ0(t), t∈J1 is absolutely continuous and the functionϕ˙0(t)is bounded; 2.2. there exist compact sets K0 ⊂ O and U0 ⊂ V containing neighborhoods of sets ϕ0(J1)∪x0([t00, t10])andclu0(J), respectively, such that the functionf(t, x, y, u)is bounded on the set J×K2
0 ×U0; 2.3. there exist the limits
lim t→t00−
˙
ϕ0(t) = ˙ϕ−0, lim w→w0
f(w, u0(t)) =f−,
wherew= (t, x, y)∈(a, t00]×O2, w0 = (t00, ϕ0(t00), ϕ0(t00−τ0)).Then there exist numbers ε2 ∈(0, ε1]and δ2 ∈(0, δ1]such that
max t∈[ˆτ ,t10+δ2]
|∆x(t)|≤O(εδµ)1 (2.6)
for arbitrary(ε, δµ)∈[0, ε2]×V−,where V−={δµ∈V :δt
0 ≤0, δτ ≤0}.Moreover, ∆x(t00) =ε
h
δϕ(t00) +{ϕ˙−0 −f−}δt0
i
+o(εδµ). (2.7)
Lemma 2.3. Let the conditions 2.1 and 2.2 of Lemma 2.2 hold, and there exist the limits
lim t→t00+
˙
ϕ0(t) = ˙ϕ+0, lim w→w0
f(w, u0(t)) =f+, w∈[t00, b)×O2.
Then there exist numbersε2∈(0, ε1]andδ2∈(0, δ1]such that the inequality (2.6) is valid
for arbitrary(ε, δµ)∈[0, ε2]×V+,where V+={δµ∈V :δt0 ≥0, δτ ≥0}.Moreover, ∆x(t00+εδt0) =ε
h
δϕ(t00) +{ϕ˙+0 −f+}δt0
i
+o(εδµ). (2.8)
Lemmas 2.2 and 2.3 can be proved in analogy to Lemma 2.4 (see [11]).
1Here and throughout the following, the symbols O(t;εδµ), o(t;εδµ) stand for quantities (scalar or vector) that have the corresponding order of smallness with respect toεuniformly with respect totand
3. Formulation of main results
Theorem 3.1. Let the conditions of Lemma 2.2 hold. Then there exist numbers ε2 ∈ (0, ε1] andδ2∈(0, δ1]such that
∆x(t;εδµ) =εδx(t;δµ) +o(t;εδµ) (3.1)
for arbitrary(t, ε, δµ)∈[t00, t10+δ2]×[0, ε2]×V− and
δx(t;δµ) =Y(t00;t){ϕ˙0−−f−}δt0+β(t;δµ), (3.2) where
β(t;δµ) =Y(t00;t)δϕ(t00) +
Z t00
t00−τ0
Y(ξ+τ0;t)fy[ξ+τ0]δϕ(ξ)dξ
−
n Z t
t00
Y(ξ;t)fy[ξ] ˙x0(ξ−τ0)dξ
o
δτ+
Z t
t00
Y(ξ;t)fu[ξ]δu(ξ)dξ. (3.3)
HereY(ξ;t) is then×n-matrix function satisfying the linear functional-differential equa-tion with advanced argument
Yξ(ξ;t) =−Y(ξ;t)fx[ξ]−Y(ξ+τ0;t)fy[ξ+τ0], ξ ∈[t00, t], (3.4)
and the condition
Y(ξ;t) =
(
I f or ξ=t,
Θf or ξ > t, (3.5)
fx= ∂x∂ f, fx[ξ] =fx(ξ, x0(ξ), x0(ξ−τ0), u0(ξ)); I is the identity matrix and Θis the zero matrix.
Some comments.The expression (3.2) is called the variation formula.
c1. Theorem 3.1 corresponds to the case when the variations at the pointst00 and τ0 are performed simultaneously on the left.
c2. The summand
−
n Z t
t00
Y(ξ;t)fy[ξ] ˙x0(ξ−τ0)dξ
o
δτ
in formula (3.3) is the effect of perturbation of the delayτ0.
c3. The expression
Y(t00;t){ϕ˙−0 −f−}δt0
is the effect of continuous initial condition (2.2) and perturbation of the initial moment t00.
c4. The expression
Y(t00;t)δϕ(t00) +
Z t00
t00−τ0
Y(ξ+τ0;t)fy[ξ+τ0]δϕ(ξ)dξ+
Z t
t00
Y(ξ;t)fu[ξ]δu(ξ)dξ
in formula (3.3) is the effect of perturbations of initialϕ0(t) and controlu0(t) functions.
c5. The variation formula allow one to obtain an approximate solution of the perturbed functional-differential equation
˙
with the perturbed initial conditionx(t) =ϕ0(t) +εδϕ(t), t∈[ˆτ , t00+εδt0].In fact, for a sufficiently smallε∈(0, ε2] from (3.1) it follows x(t)≈x0(t) +εδx(t;δµ) (see (2.5)).
c6. In the variation formula proved in [12], for the equation (1.1) instead of the expression
Z t
t00
Y(ξ;t)fu[ξ]δu(ξ)dξ
(see (3.3)), we have Z t
t00
Y(ξ;t)δf(ξ, x0(ξ), x0(ξ−τ0)dξ,
whereδf is a perturbation of the right-hand side of the equation (1.1). The local version of the formula (3.2) follows from the formula obtained in [12] if the functionf additionally satisfies the conditions : the function fu(t, x, y, u0(t)) is continuously differentiable with respect toxand y; there exists the limit
lim w→w01
f(w, u0(t)), w= (t, x, y)∈(a, t00+τ0]×O2
wherew01= (t00+τ0, x0(t00+τ0), ϕ0(t00)),with t00+τ0 < t10.In the present work vari-ation formulas are proved without of these conditions.
Theorem 3.2. Let the conditions of Lemma 2.3 hold. Then for eachˆt0 ∈(t00, t10) there
exist numbersε2 ∈(0, ε1]andδ2∈(0, δ1]such that for arbitrary (t, ε, δµ)∈[ˆt0, t10+δ2]× [0, ε2]×V+,formula (3.1) holds and
δx(t;δµ) =Y(t00;t){ϕ˙0+−f+}δt0+β(t;δµ). (3.6) Theorem 3.2 corresponds to the case when the variations at the points t00 and τ0 are performed simultaneously on the right. Theorems 3.1 and 3.2 are proved by a scheme given in [3]. The following assertion is a corollary to Theorems 3.1 and 3.2.
Theorem 3.3. Let the assumptions of Theorems 3.1 and 3.2 be fulfilled. Moreover,
˙
ϕ−0 −f−= ˙ϕ+
0 −f+=: ˆf .Then for each ˆt0 ∈(t00, t10) there exist numbersε2∈(0, ε1]and δ2∈(0, δ1]such that for arbitrary(t, ε, δµ)∈[ˆt0, t10+δ2]×[0, ε2]×V,formula (3.1) holds,
where
δx(t;δµ) =Y(t00;t) ˆf δt0+β(t;δµ).
Theorem 3.3 corresponds to the case when at the points t00 and τ0 two-sided varia-tions are simultaneously performed. All assumpvaria-tions of Theorem 2.3 are satisfied if the function f(t, x, y, u) is continuous, the function ϕ0(t) is continuously differentiable and the function u0(t) is continuous at the point t00. Clearly, in this case ˆf = ˙ϕ0(t00) − f(t00, ϕ0(t00), ϕ0(t00−τ0), u0(t00)).
4. Proof of theorem 3.1
Here and in what follows we shall assume that t0 = t00+εδt0, τ = τ0 +εδτ, ϕ(t) = ϕ0(t) +εδϕ(t), u(t) =u0(t) +εδu(t).Letε2 ∈(0, ε1] be so small (see Lemma 2.2) that for arbitrary (ε, δµ) ∈ (0, ε2]×V− the following inequalities hold t00−τ ≤ t0, t0+τ ≥t00. The function ∆x(t) (see (2.5)) satisfies the equation
˙
=fx[t]∆x(t) +fy[t]∆x(t−τ0) +εfu[t]δu(t) +r(t;εδµ) (4.1) on the interval [t00, t10+δ2],where
r(t;εδµ) =f(t, x0(t) + ∆x(t), x0(t−τ) + ∆x(t−τ), u(t))−f[t]
−fx[t]∆x(t)−fy[t]∆x(t−τ0)−εfu[t]δu(t), (4.2) f[t] =f(t, x0(t), x0(t−τ0), u0(t)).
By using the Cauchy formula ([3],p.21), one can represent the solution of equation (4.1) in the form
∆x(t) =Y(t00;t)∆x(t00) +ε
Z t
t00
Y(ξ;t)fu[ξ]δu(ξ)dξ
+ 1
X
i=0
Ri(t;t00, εδµ), t∈[t00, t10+δ2], (4.3)
where
R0(t;t00, εδµ) =
Rt00
t00−τ0Y(ξ+τ0;t)fy[ξ+τ0]∆x(ξ)dξ,
R1(t;t00, εδµ) =
Rt
t00Y(ξ;t)r(ξ;εδµ)dξ
(4.4)
and Y(ξ;t) is the matrix function satisfying equation (3.4) and condition (3.5). The functionY(ξ;t) is continuous on the set Π ={(ξ, t) : a≤ξ≤t, t∈J}([3], Lemma 2.1.7). Therefore,
Y(t00;t)∆x(t00) =εY(t00;t)
h
δϕ(t00) +{ϕ˙−0 −f−}δt0
i
+o(t;εδµ) (4.5)
(see (2.7)). One can readily see that
R0(t;t00, εδµ) =ε
Z t0
t00−τ0
Y(ξ+τ0;t)fy[ξ+τ0]δϕ(ξ)dξ
+
Z t00
t0
Y(ξ+τ0;t)fy[ξ+τ0]∆x(ξ)dξ=ε
Z t00
t00−τ0
Y(ξ+τ0;t)fy[ξ+τ0]δϕ(ξ)dξ
+o(t;εδµ) (4.6)
(see (2.5) and (2.6)). We introduce the notations:
f[t;s, εδµ] =f(t, x0(t) +s∆x(t), x0(t−τ0) +s{x0(t−τ)−x0(t−τ0) +∆x(t−τ)}, u0(t) +sεδu(t)), σ(t;s, εδµ) =fx[t;s, εδµ]−fx[t], ρ(t;s, εδµ) =fy[t;s, εδµ]−fy[t], θ(t;s, εδµ) =fu[t;s, εδµ]−fu[t]. It is easy to see that
=
Z 1
0 d
dsf[t;s, εδµ]ds=
Z 1
0
n
fx[t;s, εδµ]∆x(t) +fy[t;s, εδµ]{x0(t−τ)
−x0(t−τ0) + ∆x(t−τ)}+εfu[t;s, εδµ]δu(t)
o
ds
=
h Z 1 0
σ(t;s, εδµ)ds
i
∆x(t) +
h Z 1 0
ρ(t;s, εδµ)ds
i
{x0(t−τ)
−x0(t−τ0) + ∆x(t−τ)}+ε
h Z 1 0
θ(t;s, εδµ)ds
i
δu(t)
+fx[t]∆x(t) +fy[t]{x0(t−τ)−x0(t−τ0) + ∆x(t−τ)}+εfu[t]δu(t).
By taking account of last relation fort∈[t00, t10+δ2] we have R1(t;t00, εδµ) =
6
X
i=2
Ri(t;t00, εδµ), where
R2(t;t00, εδµ) =
Z t
t00
Y(ξ;t)σ1(ξ;εδµ)∆x(ξ)dξ,
σ1(ξ;εδµ) =
Z 1
0
σ(ξ;s, εδµ)ds, R3(t;t00, εδµ)
=
Z t
t00
Y(ξ;t)ρ1(ξ;εδµ){x0(ξ−τ)−x0(ξ−τ0) + ∆x(ξ−τ)}dξ,
ρ1(ξ;εδµ) =
Z 1
0
ρ(ξ;s, εδµ)ds, R4(t;t00, εδµ)
=ε
Z t
t00
Y(ξ;t)θ1(ξ;εδµ)δu(ξ)dξ, θ1(ξ;εδµ)
=
Z 1
0
θ(ξ;s, εδµ)ds, R5(t;t00, εδµ) =
Z t
t00
Y(ξ;t)fy[ξ]{x0(ξ−τ)
−x0(ξ−τ0)}dξ, R6(t;t00, εδµ) =
Z t
t00
Y(ξ;t)fy[ξ]{∆x(ξ−τ)
−∆x(ξ−τ0)}dξ
(see (4.2)). The functionx0(t), t∈[ˆτ , t10+δ2] is absolutely continuous, then for each fixed Lebesgue pointξ ∈(t00, t10+δ2) of function ˙x0(ξ−τ0) we get
x0(ξ−τ)−x0(ξ−τ0) =
Z ξ−εδτ
ξ ˙
x0(s−τ0)ds
=−εx˙0(ξ−τ0)δτ+γ(ξ;εδµ), (4.7) with
lim ε→0
γ(ξ;εδµ)
ε = 0 uniformly forδµ∈V
Thus, (4.8) is valid for almost all points of the interval (t00, t10+δ2). From (4.7) taking into account boundedness of the function
˙ x0(t) =
(
˙
ϕ0(t), t∈[ˆτ , t00], f[t], t∈(t00, t10+δ2] it follows
|x0(ξ−τ)−x0(ξ−τ0)|≤O(εδµ) and
¯ ¯
¯γ(ξ;εδµ)
ε
¯ ¯
¯≤const. (4.9) It is clear that
|∆x(ξ−τ)−∆x(ξ−τ0)|=
(
o(ξ;εδµ) forξ∈[t00, t0+τ], O(ξ;εδµ) forξ∈[t0+τ, t00+τ0]
(4.10)
(see (2.6)). We note that there existsL(·)∈L(J,[0,∞)) such that
|f(t, x1, y1, u1)−f(t, x2, y2, u2)|≤L(t)(|x1−x2|+|y1−y2 |+|u1−u2 |), t∈J,(xi, yi, ui)∈K02×U0, i= 1,2 ([3], Lemma 2.1.2).
Letξ∈[t00+τ0, t10+δ2] thenξ−τ ≥t00, ξ−τ0≥t00,therefore
|∆x(ξ−τ)−∆x(ξ−τ0)|≤
Z ξ−τ
ξ−τ0
|∆x(s)˙ |ds
≤
Z ξ−τ
ξ−τ0
L(s)
n
|∆x(s)|+|x0(s−τ)−x0(s−τ0)| +|∆x(s−τ)|+ε|δu(s)|
o
ds=o(ξ;εδµ) (4.11) (see (4.1),(2.4),(2.6) and (4.9)). It is not difficult to see that for the expressionsRi(t;t00, εδµ), i= 2,6 we get
|R2(t;t00, εδµ)|≤kY kO(εδµ)σ2(εδµ), |R3(t;t00, εδµ)|
≤kY kO(εδµ)ρ2(εδµ), |R4(t;t00, εδµ)|≤εαkY kθ2(εδµ), R5(t;t00, εδµ)
=−ε
h Z t
t00
Y(ξ;t)fy[ξ] ˙x0(ξ−τ0)dξ
i
δτ +γ1(t;εδµ),|R6(t;t00, εδµ)|
≤kY k
Z t10+δ2
t00
|fy[ξ]||∆x(ξ−τ)−∆x(ξ−τ0)|dξ=o(εδµ) (4.12) (see (2.6),(2.3),(4.7) and (4.9)-(4.11)). Here
kY k= sup
n
|Y(ξ;t)|: (ξ, t)∈Π
o
, γ1(t;εδµ) =
Z t
t00
Y(ξ;t)fy[ξ]γ(ξ;εδµ)dξ
σ2(εδµ) =
Z t10+δ2
t00
|σ1(ξ;εδµ)|dξ, ρ2(εδµ) =
Z t10+δ2
t00
θ2(εδµ) =
Z t10+δ2
t00
nXk
i=1
|δui(ξ)|
o
|θ1(ξ;εδµ)|dξ, Obviously,
¯ ¯
¯γ1(t;εδµ)
ε
¯ ¯
¯≤kY k
Z t10+δ2
t00
|fy[ξ]|
¯ ¯ ¯γ(ξ;εδµ) ε ¯ ¯ ¯dξ.
By the Lebesguer theorem on passing to the limit under the integral sign, we have
lim
ε→0σ2(εδµ) = 0, εlim→0ρ2(εδµ) = 0, εlim→0θ2(εδµ) = 0, εlim→0
¯ ¯
¯γ1(t;εδµ)
ε
¯ ¯ ¯= 0
uniformly for (t, δµ)∈[t00, t10+δ2]×V− (see(4.9)). Thus,
Ri(t;t00, εδµ) =o(t;εδµ), i= 2,3,4; (4.13) and
R5(t;t00, εδµ) =−ε
h Z t
t00
Y(ξ;t)fy[ξ] ˙x0(ξ−τ0)dξ
i
δτ +o(t;εδµ). (4.14)
On the basis of (4.12)-(4.14) we obtain
R1(t;t00, εδµ) =−ε
h Z t
t00
Y(ξ;t)fy[ξ] ˙x0(ξ−τ0)dξ
i
δτ +o(t;εδµ). (4.15)
From (4.3) by virtue of (4.5),(4.6) and (4.15) we obtain (3.1), where δx(t;δµ) has form (3.2).
5. Proof of theorem 3.2
Let ˆt0 ∈(t00, t10) be a fixed point, and letε2 ∈(0, ε1] be so small (see Lemma 2.3) that t0 <tˆ0 and t0−τ0< t00,for arbitrary (ε, δµ)∈(0, ε1]×V+.The function ∆x(t) satisfies equation (4.1) on the interval [t0, t10+δ2].By using the Cauchy formula, we can represent it in the form
∆x(t) =Y(t0;t)∆x(t0) +ε
Z t
t0
Y(ξ;t)fu[ξ]δu(ξ)dξ+ 1
X
i=0
Ri(t;t0, εδµ), (5.1) (see (4.4)). The matrix function Y(ξ;t) is continuous on [t00,ˆt0]×[ˆt0, t10+δ2] ⊂ Π, therefore
Y(t0;t)∆x(t0) =εY(t00;t)[δϕ(t00) + ( ˙ϕ+0 −f+)]δt0+o(εδµ) (5.2) (see (2.8)). Now let us transformR0(t;t0, εδµ).It is not difficult to see that
R0(t;t0, εδµ) =ε
Z t00
t0−τ0
Y(ξ+τ0;t)fy[ξ+τ0]δϕ(ξ)dξ
+
Z t0
t00
Y(ξ+τ0;t)fy[ξ+τ0]∆x(ξ)dξ
=ε
Z t00
t00−τ0
Y(ξ+τ0;t)fy[ξ+τ0]δϕ(ξ)dξ+o(t;εδµ). (5.3) In a similar way, fort∈[ˆt0, t10+δ2] one can prove
R1(t;t0, εδµ) =−ε
h Z t
t00
Y(ξ;t)fy[ξ] ˙x0(ξ−τ0)dξ
i
Finally, we note that
ε
Z t
t0
Y(ξ;t)fu[ξ]δu(ξ)dξ=ε
Z t
t00
Y(ξ;t)fu[ξ]δu(ξ)dξ+o(t;εδµ). (5.5)
fort ∈ [ˆt0, t10+δ2]. By taking account of (5.2)-(5.5), from (5.1), we obtain (3.1), where δx(t;εδµ) has form (3.6).
Acknowledgement. The work was supported by the Sh.Rustaveli National Science Foundation (Georgia), Grant No.GNSF/ST08/3-399 and Centre National de la Recherche Scientifique CNRS (France), PICS-4962.
References
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Tamaz Tadumadze was born in the village Khunjulauri, Samtredya re-gion, Georgia, 1947. Graduated with Honors from Tbilisi State University (TSU) in 1970. He received Ph.D. and Dr. Sc. degrees in mathematics from TSU (1975) and A. Razmadze Mathematical Institute (1985), respec-tively. Head of Differential equations and control theory division of I. Vekua Institute of Applied Mathematics of TSU (1993-2006). Currently he is full professor of TSU. A. Razmadze Award of the Georgian National Academy of Sciences (2001). Member of the Georgian Mathematical Union. His re-search interests include Variation Formulas, Well-Posednisse, Inverse Prob-lems for Differential Equations; Optimal Control of Functional-Differential Equations (Necessary Conditions, Existence Theorems, Regular Perturbations and Well-Posednisse, Numerical Methods).
